# The GMM estimator is reliable under the given conditions

Consider the linear model

$$y_t= x_t'\beta +\epsilon_t$$ for $$t=1,...,T$$.

where $$x_t= ( x_{1t} \ \ x_{2t} \ \ ... \ \ x_{kt})$$ and $$\beta$$ is $$(k\times 1)$$ vector of unknown coefficients.

Given $$z_t= ( z_{1t} \ \ z_{2t} \ \ ... \ \ z_{ht})'$$ with $$E(z_t) \not= 0$$ for $$h>k$$ is a white noise disturbance term

with $$E(z_t\epsilon_t)=0 \ \ \forall t$$.

Also, a symmetric and positive definite weight matrix $$S_n$$ is defined.

Firstly, we need to find the Generalized Method of Moments (GMM) estimator of $$\beta$$.

Secondly, we need to show that $$\hat{\beta}_{GMM}$$ is a consistent estimator of $$\beta$$ for $$plim(\frac{1}{T} \sum z_t x_t')= R$$ and $$plim(S_n)=S$$ for constant and finite matrices $$R$$ and $$S$$.

Thirdly, we need to show the reliability of this estimator $$\hat{\beta}_{GMM}$$ (with mathematical expressions and calculations) when the linear model is misspecified with $$\epsilon_t = \gamma T^{-1/2} + e_t$$ where $$e_t$$ is iid disturbance with $$E(e_t)=0$$

What I did is that...

I did the first and second parts.

I find the Generalized Method of Moments (GMM) estimator of $$\beta$$ as

$$\hat{\beta}_{GMM} = (X'ZS_nZ'X)^{-1}X'ZS_nZ'y$$

As for the second part,

$$\hat{\beta}_{GMM} = (X'ZS_nZ'X)^{-1}X'ZS_nZ'(X\beta +\epsilon)$$

$$\hat{\beta}_{GMM} = \beta +(X'ZS_nZ'X)^{-1}X'ZS_nZ'\epsilon$$

$$plim(\hat{\beta}_{GMM}) = \beta +(plim(\frac{X'Z}{T})plim(S_n)plim(\frac{Z'X}{T}))^{-1}plim(\frac{X'Z}{T})plim(S_n)plim(\frac{Z'\epsilon}{T})$$

$$plim(\hat{\beta}_{GMM}) = \beta + (R'SR)^{-1}R'S plim(\frac{Z'\epsilon}{T})$$

since $$plim(\frac{Z'\epsilon}{T})=0$$ by law of large numbers, $$plim(\hat{\beta}_{GMM}) = \beta$$

Mu actual question is the third part in the gray box. I could not show whether $$\hat{\beta}_{GMM}$$ is reliable or not under the given conditions.

I think that the GMM estimator is reliable if the estimator is consistent and efficient. But I could not proceed this. And I could not show this third part.

what I did for the part 3

Population moment condition requires $$E(z_t\epsilon_t)=0$$

If this requirement holds, then the estimator $$\hat{\beta}_{GMM}$$ will be reliable.

We know that $$E(z_t)=\mu_z\not= 0$$

$$E(z_t \epsilon_t)=E(z_t(\gamma T^{-1/2}z_t z_te_t))$$

$$= \gamma T^{-1/2}E(z_t)+E( z_te_t)= \gamma T^{-1/2}\mu_z + E( E(z_te_t|z_t))= \gamma T^{-1/2}\mu_z + E(z_t E(e_t|z_t))= \gamma T^{-1/2}\mu_z +0$$

where $$E(e_t|z_t)=0$$ since $$e_t$$ is i.i.d and $$E(e_t)=0$$

So, $$E(z_t \epsilon_t)= \gamma T^{-1/2}\mu_z\not=0$$ for $$\gamma\not=0$$. In this case, the GMM estimation is not reliable. On the other hand, if $$\gamma=0$$, then the GMM estimator becomes reliable.

What do you think about this solution? Or what is your idea on how to show this question? I will be glad if you help me. Thank you.